Cook the potatoes: Place the potatoes in a large pot of salted water and bring to boil over high heat. Turn the flame down to medium-low and simmer the potatoes until tender, about 10 minutes. Drain the potatoes and return them to the pot. Add the butter, salt, and half of the milk and mash with a fork or potato masher until the potatoes are broken down. Pour in the remaining milk and continue to stir until very smooth and lump-free. (Note: potatoes can be made up to two days in advance. Reheat and add a little extra milk to soften them so they are spreadable.)

Cook the meat: Preheat the oven to 350 degrees F. In a large oven-proof skillet, saute the onion and carrots over medium flame until soft, 5 minutes. Push them to the side and crank the heat to high. Add the beef in the center and break apart with your spatula. When the meat is cooked through and thoroughly broken down, mix it together with the vegetables. Continue to cook, stirring occasionally, until the meat is browned, about 5 more minutes. Add the garlic, salt, paprika, and cayenne and continue to saute on high heat until the meat is very brown, another 5 minutes. Stir in the flour until incorporated. Then add the tomato paste and the broth, scraping up the brown bits from the bottom of the pan. cook, stirring occasionally, until the broth has reduced by at least half.

Assemble the pie: Spoon the potatoes over the meat mixture and spread with your spatula to form an even layer. Sprinkle the cheese over the top and bake in the oven for 20 minutes, until the pie is bubbling and the cheese is melted.

Finish the pie under the broiler until the top is brown and crusty, about 2 minutes.